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Properties

Label 1-3024-3024.1861-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.435 + 0.900i$
Analytic cond. $324.973$
Root an. cond. $324.973$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.984 − 0.173i)5-s + (−0.984 + 0.173i)11-s + (−0.342 − 0.939i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.342 + 0.939i)29-s + (−0.766 − 0.642i)31-s + (−0.866 − 0.5i)37-s + (−0.939 + 0.342i)41-s + (−0.984 + 0.173i)43-s + (−0.766 + 0.642i)47-s + i·53-s + 55-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)5-s + (−0.984 + 0.173i)11-s + (−0.342 − 0.939i)13-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.342 + 0.939i)29-s + (−0.766 − 0.642i)31-s + (−0.866 − 0.5i)37-s + (−0.939 + 0.342i)41-s + (−0.984 + 0.173i)43-s + (−0.766 + 0.642i)47-s + i·53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.435 + 0.900i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1861, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.435 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1390667268 + 0.08718003235i\)
\(L(\frac12)\) \(\approx\) \(0.1390667268 + 0.08718003235i\)
\(L(1)\) \(\approx\) \(0.6619629611 - 0.1333241657i\)
\(L(1)\) \(\approx\) \(0.6619629611 - 0.1333241657i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.984 - 0.173i)T \)
11 \( 1 + (-0.984 + 0.173i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.342 + 0.939i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.984 + 0.173i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + iT \)
59 \( 1 + (-0.984 - 0.173i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (0.342 + 0.939i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.342 - 0.939i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.68738363264891143379763426021, −18.348470848950065691533111131752, −17.26034503102718272065615111912, −16.569245658453666073581774355, −15.908667120923162900864199054037, −15.364117295920944555222807699450, −14.58566164125607976971400862136, −13.902411088992446412520990570351, −13.109908710916291335528811817299, −12.20428204649229915275109872067, −11.77720881108522477751360683369, −11.02845865270778809199640391756, −10.1901441606514374580366340615, −9.61745473819656247294660615809, −8.41732571006211012221214505512, −8.05442883609705554015873125340, −7.25907289861353243359769271875, −6.56721711571109443391268694685, −5.47831563490767369792977540654, −4.910521794530126881458997833506, −3.71321943483761227438131200447, −3.49852710736560967524154953310, −2.26073620981644381113298550879, −1.402799073728722760332382774390, −0.05253194814946321977425835783, 0.43987030586424915803804208615, 1.64166631852505918699183330955, 2.93806020770684907350933951579, 3.22915572060077249347293300070, 4.421993670017722677799541831137, 5.098535110029338230279905147853, 5.68388478572666535503015430274, 7.00050961990099942786951684464, 7.5161290563273660118626065080, 8.084608284687007322419496425190, 8.89572489339800207830043701363, 9.822474554614170411674418518096, 10.48838416033613486772776629252, 11.260468394500344778322217402982, 11.97539363542168178150004181215, 12.61941042315430170085477831611, 13.23892915768308491583402362539, 14.15962588929949119337975440576, 14.94253423223176091089866102130, 15.55376965633569866177335439509, 16.16634398764757688433529632228, 16.69061106988324967998733552481, 17.823113981491556965259376270366, 18.30239018023247251150070641040, 18.902649181434761896682521847043

Graph of the $Z$-function along the critical line